NAME

       zbdsqr.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zbdsqr (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, RWORK, INFO)
           ZBDSQR

Function/Subroutine Documentation

   subroutine zbdsqr (characterUPLO, integerN, integerNCVT, integerNRU, integerNCC, double
       precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension(
       ldvt, * )VT, integerLDVT, complex*16, dimension( ldu, * )U, integerLDU, complex*16,
       dimension( ldc, * )C, integerLDC, double precision, dimension( * )RWORK, integerINFO)
       ZBDSQR

       Purpose:

            ZBDSQR computes the singular values and, optionally, the right and/or
            left singular vectors from the singular value decomposition (SVD) of
            a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
            zero-shift QR algorithm.  The SVD of B has the form

               B = Q * S * P**H

            where S is the diagonal matrix of singular values, Q is an orthogonal
            matrix of left singular vectors, and P is an orthogonal matrix of
            right singular vectors.  If left singular vectors are requested, this
            subroutine actually returns U*Q instead of Q, and, if right singular
            vectors are requested, this subroutine returns P**H*VT instead of
            P**H, for given complex input matrices U and VT.  When U and VT are
            the unitary matrices that reduce a general matrix A to bidiagonal
            form: A = U*B*VT, as computed by ZGEBRD, then

               A = (U*Q) * S * (P**H*VT)

            is the SVD of A.  Optionally, the subroutine may also compute Q**H*C
            for a given complex input matrix C.

            See "Computing  Small Singular Values of Bidiagonal Matrices With
            Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
            LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
            no. 5, pp. 873-912, Sept 1990) and
            "Accurate singular values and differential qd algorithms," by
            B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
            Department, University of California at Berkeley, July 1992
            for a detailed description of the algorithm.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  B is upper bidiagonal;
                     = 'L':  B is lower bidiagonal.

           N

                     N is INTEGER
                     The order of the matrix B.  N >= 0.

           NCVT

                     NCVT is INTEGER
                     The number of columns of the matrix VT. NCVT >= 0.

           NRU

                     NRU is INTEGER
                     The number of rows of the matrix U. NRU >= 0.

           NCC

                     NCC is INTEGER
                     The number of columns of the matrix C. NCC >= 0.

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the n diagonal elements of the bidiagonal matrix B.
                     On exit, if INFO=0, the singular values of B in decreasing
                     order.

           E

                     E is DOUBLE PRECISION array, dimension (N-1)
                     On entry, the N-1 offdiagonal elements of the bidiagonal
                     matrix B.
                     On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
                     will contain the diagonal and superdiagonal elements of a
                     bidiagonal matrix orthogonally equivalent to the one given
                     as input.

           VT

                     VT is COMPLEX*16 array, dimension (LDVT, NCVT)
                     On entry, an N-by-NCVT matrix VT.
                     On exit, VT is overwritten by P**H * VT.
                     Not referenced if NCVT = 0.

           LDVT

                     LDVT is INTEGER
                     The leading dimension of the array VT.
                     LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

           U

                     U is COMPLEX*16 array, dimension (LDU, N)
                     On entry, an NRU-by-N matrix U.
                     On exit, U is overwritten by U * Q.
                     Not referenced if NRU = 0.

           LDU

                     LDU is INTEGER
                     The leading dimension of the array U.  LDU >= max(1,NRU).

           C

                     C is COMPLEX*16 array, dimension (LDC, NCC)
                     On entry, an N-by-NCC matrix C.
                     On exit, C is overwritten by Q**H * C.
                     Not referenced if NCC = 0.

           LDC

                     LDC is INTEGER
                     The leading dimension of the array C.
                     LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (2*N)
                     if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  If INFO = -i, the i-th argument had an illegal value
                     > 0:  the algorithm did not converge; D and E contain the
                           elements of a bidiagonal matrix which is orthogonally
                           similar to the input matrix B;  if INFO = i, i
                           elements of E have not converged to zero.

       Internal Parameters:

             TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
                     TOLMUL controls the convergence criterion of the QR loop.
                     If it is positive, TOLMUL*EPS is the desired relative
                        precision in the computed singular values.
                     If it is negative, abs(TOLMUL*EPS*sigma_max) is the
                        desired absolute accuracy in the computed singular
                        values (corresponds to relative accuracy
                        abs(TOLMUL*EPS) in the largest singular value.
                     abs(TOLMUL) should be between 1 and 1/EPS, and preferably
                        between 10 (for fast convergence) and .1/EPS
                        (for there to be some accuracy in the results).
                     Default is to lose at either one eighth or 2 of the
                        available decimal digits in each computed singular value
                        (whichever is smaller).

             MAXITR  INTEGER, default = 6
                     MAXITR controls the maximum number of passes of the
                     algorithm through its inner loop. The algorithms stops
                     (and so fails to converge) if the number of passes
                     through the inner loop exceeds MAXITR*N**2.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       Definition at line 223 of file zbdsqr.f.

Author

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